Table Of ContentSeparability and Fourier representations of density matrices
Arthur O. Pittenger and Morton H. Rubin
‡
† ‡
†Department of Mathematics and Statistics, University of Maryland, Baltimore County, Baltimore, MD 21228-5398
‡Department of Physics, University of Maryland, Baltimore County, Baltimore, MD 21228-5398
(January 5, 2000)
Using the finite Fourier transform, we introduce a generalization of Pauli-spin matrices for d-
dimensional spaces, and the resulting set of unitary matrices S(d) is a basis for d×d matrices. If
N = d1 ×d2 ×···×db and H[N] = H[dk], we give a sufficient condition for separability of a
density matrix ρ relative to the H[dk] in terms of the L1 norm of the spin coefficients of ρ. Since
N
the spin representation depends on the form of the tensor product, the theory applies to both full
andpartialseparabilityonagivenspaceH[N]. Itfollowsfrom thisresultthatforaprescribedform
of separability, there is always a neighborhood of the normalized identity in which every density
0 matrix isseparable. Wealso showthatfor everyprime pandn>1thegeneralized Wernerdensity
0 matrix W[pn](s) is fully separable if and only if s≤ 1+pn−1 −1.
0
2 03.67.Lx, 03.67.Hk, 03.65.Ca (cid:0) (cid:1)
n
a
J
7 I. INTRODUCTION
1
v One of the predictions of quantum mechanics is that spatially separatedcomponents of a systemcan be entangled.
4 The consequent prediction of non-classical correlations among the separated components of a quantum system has
1 led to critiques of the foundations of quantum mechanics, as in the famous Einstein, Podolsky, Rosen paper [1],
0 and to experiments that have confirmed the predicted non-classical correlations, as in [2]. Interest in entangled
1 systems has been heightened by proposed applications in quantum computation, for example [3], and in quantum
0
communication,asexemplifiedmostdramaticallyby teleportation[4]. As a resultthere havebeenmanypublications
0
which have examined various aspects of entanglement, its measurement, and its use in quantum communication such
0
/ as references [5–9] to mention only a few recent papers.
h
In this paper we shall be interested in the separability properties of quantum systems in states defined on finite
p
dimensional Hilbert spaces H = H H , where the H denote the Hilbert spaces of the subsystems. A state
- 1⊗···⊗ n k
t specifiedbyadensitymatrixρissaidtobecompletelyseparableonH ifitisaconvexcombinationoftensorproducts:
n
a
u ρ= p(a)ρ(1)(a) ρ(n)(a), (1)
⊗···⊗
q a
X
:
v
where ρ(k)(a) is a density matrix on H . Since the same ρ can have different convex representations, it has proven
i k
X difficult to determine generally applicable operational conditions for separability, and determining such conditions is
r one of the motivations for this paper. It is also possible to have different types of separability by allowing sets of the
a subsystems to be entangled, cf. [9,10], and one can describe a lattice of levels of separability. The theory we develop
here applies to all of these various definitions of separability.
A necessary condition for separability is that the partial transpose ρTr of a state ρ should be a state [11]. If we
represent ρ as a matrix, this means that if ρ=(ρ ) then (taking r =1)
j1...jn,k1...kn,
ρT1 = ρ (2)
k1j2...jn,j1k2...kn,
is also a density matrix. It is easy to confirm tha(cid:0)t if a density ma(cid:1)trix is separable, its partial transposes are also
separable, but it has been shown [12] that the converse is true only in the 2 2 and 2 3 cases. In the proof of
⊗ ⊗
this last result [12], a necessary and sufficient criterion for separability was established, but there seems to be no
operationalway of using this criterionas a generaltool. Other studies of separability,such as those in [9,13–15]have
found operationally useful necessary conditions and sufficient conditions for classes of densities or for special cases,
but no general sufficient conditions with a breadth of applicability analogous to that of the Peres condition.
‡Present address: The Centre for Quantum Computation, Clarendon Laboratory, Oxford University
1
Broadly speaking, necessary conditions tend to be described in the computational basis while sufficient conditions
for 2-level systems tend to be described in terms of the Pauli spin basis. That observation motivated the derivation
of a change-of-basisformula in [16] which facilitates the strategy of checking whether necessary conditions derived in
the computational basis are sufficient by using the (real) Pauli spin basis. This approach leads to general sufficient
conditions for full separability which essentially give the condition in [14] as a corollary and also leads to necessary
and sufficient conditions for full separability of a parametrized family of n-qubit densities which all satisfy the Peres
condition. The difficulty with extending this approach to d-level systems is that the generally accepted definition of
spin matrices as generators of rotations does not capture the computationally useful features of the Pauli matrices
when d 3. One of the basic purposes of this paper is to propose a general definition of d-level spin matrices which
≥
possess many of those computational properties.
The Pauli matrices are special in that they are both Hermitian and unitary, and together with the identity matrix
σ they form a basis of the set of 2 2 matrices. Our strategy is to generalize the role of the Pauli matrices as
0
×
a basis of unitary matrices at the expense of Hermiticity. We show the applicability of these proposed d-level spin
matrices and in the appendix examine the d=3 case in some detail, identifying properties analogous to those of the
Paulimatrices. We also define directly a generalcharacterizationof certain classes of trace one projections of d–level
systems. We use those projections to establish necessary and sufficient conditions for full separability of generalized
Werner densities composed of any number n of d–dimensional subsystems for any prime d. In addition, we establish
a general sufficient condition for full or partial separability of densities of any dimension. Analogous results for full
separability were obtained in [16] for d=2 by essentially the same methodology.
Otherauthors[17]haveusedadifferentsetofoperatorsinthed=3caseandsomeseparabilityresultswereobtained
recently in [15]. Our proposedclass is different, and we show that stronger separability results can be obtained using
these matrices and the strategy developed in [16].
II. A NECESSARY CONDITION
As mentioned above,the Peres partial transpose condition is a generalnecessary condition for separability [11]. In
[16]aweakerbutusefulconditionwasderivedusingtheCauchy-Schwarzinequalityandhasthefollowingapplication.
Suppose j = j ...j and k = k ...k differ in each component: j = k . Let u and v be indices with u = v and
1 n 1 n r r r r
6 6
u ,v = j ,k . Then for fully separable states ρ
r r r r
{ } { }
√ρj,j√ρk,k ρu,v , (3)
≥| |
where ρ is written as a matrix in the comput(cid:0)ational basis(cid:1)defined by the tensor products of j k ,1 i n. As
i i
| ih | ≤ ≤
an application, consider the following generalization of the Werner density matrix [18] on the N = dn dimensional
Hilbert space H[N]:
1 s
W[N](s) = − I +s τ
dn ·
where I is the identity and τ is the projection defined by the state
1
ψ[N] = (0...0 + 1...1 +...+ (d 1)...(d 1) ). (4)
√d | i | i | − − i
(cid:12) E
(cid:12)
(In the sequel we let k˜ de(cid:12)note the repeated index k...k.) In the computational basis W[N](s) equals 1 s + s
j,j d−n d
when j is in k˜ :0≤k <d and equals 1d−ns otherwise. The only non-zero off-diagonal elements are W(cid:0)˜j[N,k˜](s) =(cid:1)
s. Choosing nj and k approopriately in (3), we have the necessary condition 1 s 1+dn 1 . To show that this
d ≥ −
condition is also sufficient, we will use the spin representation to prove W[N](s) is fully separable when d is prime
(cid:0) (cid:1)
and s= 1+dn−1 −1. In order to do that, however, we first need to define the spin representation.
(cid:0) (cid:1)
III. COMPUTATIONAL AND SPIN BASES
LetH[N]denoteanN-dimensionalHilbertspacewhereN =d d d . Inthissectionwedefinedifferentbases
1 2 b
for N N matrices on H[N] based on different representationso×f H[×N·]·a·s×a tensor product space, and the discussion
is pure×ly mathematical. In the applications we will be concerned with a specific representation H[N] = b H[da]
a=1
N
2
and with the corresponding separability properties of densities on H[N]. The bases used will depend on the order of
the tensor product as will the representationof a density matrix as ρ=ρ ρ , the tensor product of densities
1 b
⊗···⊗
ρk on the H[dk]. For example, we might want to examine separability of a density matrix on H[90] = H[6] H[15]
using matrices consistent with that tensor product. In a subsequent application one might want H[6] to re⊗present
the tensor product of spin 1/2 and spin 1 particles, i.e. a tensor product of H[2] and H[3], and the order of the
sub-tensor product shouldn’t affect the theory. We confirm that assertion by showing that permuting the order of
a tensor product corresponds to a conjugation operation and thus that the theory is generally applicable with only
notational changes for particular applications.
Lemma 1 Let N =d d d and suppose M is an N N matrix with M =C(1) C(b), where the
1 2 b
C(k) are d d matrices×. Let×σ··d·e×note a permutation of 1,...×,b and let M = C(σ(1)) ⊗···⊗C(σ(b)). Then for
k k σ
× { } ⊗···⊗
all such C(k)’s there is a permutation matrix Q such that Q M Q 1 =M.
σ σ σ −σ
Proof: If M(j,k)=C(1)(j ,k ) C(b)(j ,k ), the index j corresponds to the ordered b-tuple (j ,...,j ) and j
1 1 b b 1 b
·····
is uniquely defined by j = j (d d )+j (d d )+...+j and similarly for k. Let j correspond to
1 2 b 2 3 b b σ
j ,...,j and define the pe×rm··u·t×ation matrix Q×·b··y×Q (j,s)=δ(j ,s). Then Q 1 =Qt and
σ(1) σ(b) σ σ σ −σ σ
(cid:0) (cid:1)
Q M Q 1(j,k)=M (j ,k )=C(σ(1)) j ,k C(σ(b)) j ,k =M(j,k),
σ σ −σ σ σ σ σ(1) σ(1) ····· σ(b) σ(b)
completing the proof. (cid:0) (cid:1) (cid:0) (cid:1)
The motivation for this work comes from the identification of the 2 2 Hadamardmatrix as a key tool in working
with 2-level systems. Specifically, suppose ρ= 1(σ +σ ), where σ ×= m σ and σ =σ , σ =σ , and σ =σ
2 0 m m j j 1 x 2 y 3 z
are the usual Pauli matrices. Then the coefficients in the spin basis are related to those in the computational basis
by the 2 2 Hadamard matrix: P
×
1 m 1 1 ρ ρ
1 = 00 01 . (5)
m im 1 1 ρ ρ
3 2 11 10
(cid:18) − (cid:19) (cid:18) − (cid:19)(cid:18) (cid:19)
The matrices in the two bases are connected in a similar fashion:
σ σ 1 1 E E
0 1 = 0,0 0,1 , (6)
σ iσ 1 1 E E
3 2 1,1 1,0
(cid:18) (cid:19) (cid:18) − (cid:19)(cid:18) (cid:19)
where the matrices E = j k define the computational basis. Note that a systematic application of these re-
j,k
| ih |
lationships requires both the use of the real Pauli matrices and a reindexing of both the Pauli matrices and the
computational basis matrices to conform to the matrix notation.
The Hadamardmatrix isthe 2 2Fouriertransform,andweextendthe idea of(6)to d dmatrices. Theadjusted
× ×
basisA= A ,0 j,k <d isthesetofd dmatricesdefinedbyA =E ,where+denotesadditionmodulod,
j,k j,k j,j+k
{ ≤ } ×
andwedefinethe “spin”matricesS = S , 0 j,k <d usingthe analogueof(6)andthe finite Fouriertransform.
j,k
Thus (S) F (A) where F (j,k) = ex{p(2πijk≤/d)= ηjk}with η = exp(2πi/d). (We will make the dependence on d
≡ ·
explicit below.) In detail
d 1
−
S = F(j,r)A (7)
j,k r,k
r=0
X
is a sum of products of scalars times matrices. Since F is invertible, it follows that S is also a basis for the d d
×
matrices. Note that (6) is a special case of (7) with d=2 and η = 1.
−
To illustrate these ideas, it is useful to write out the results for d=3 in detail. Then η =exp(2πi/3) and
1 0 0 0 1 0 0 0 1
S = 0 1 0 S = 0 0 1 S = 1 0 0
00 01 02
0 0 1 1 0 0 0 1 0
1 0 0 0 1 0 0 0 1
S = 0 η 0 S = 0 0 η S = η 0 0
10 11 12
0 0 η2 η2 0 0 0 η2 0
1 0 0 0 1 0 0 0 1
S = 0 η2 0 S = 0 0 η2 S = η2 0 0
20 21 22
0 0 η η 0 0 0 η 0
3
The spin matrices S not only form a basis for d d matrices, but share many other properties with the real Pauli
×
matrices, which we record next. We should note that the matrices S were also defined in an earlier work by Fivel
j,k
[19] on Hamiltonians on discrete spaces, and many of the properties listed below were first established there.
Proposition 1 Fix d 2 and let S denote the corresponding set of spin matrices.
≥
(i) S is an orthogonalbasis of unitary matrices with respect to the trace inner product.
(ii) If d is odd, each matrix in S is in SU(d), while if d is even, S is in SU(d) if and only if j+k is even.
j,k
(iii) Sj,k = (S1,0)j (S0,1)k, (Sj,k)† = ηjkSd j,d k = (Sj,k)d−1 and [Sj,k,Sr,s] = ηkr ηjs Sj+r,k+s using addition
modd. · − − −
(cid:0) (cid:1)
(iv) tr(S )=0 for all (j,k)=(0,0).
j,k
6
Proof: Thekeyobservation,notedin[19],isthatthematricesaregeneratedbyS andS :S =(S )j (S )k
1,0 0,1 j,k 1,0 0,1
·
with S S =ηS . All of the remaining assertions,including orthogonality,follow from those relations and from
0,1 1,0 1,1
·
easy computations. A useful consequence of the manipulations is
(S )m =η(jk)m(m 1)/2S . (8)
j,k · − mj,mk
Unlike the Pauli matrices, these spin matrices need not be Hermitian; for example, when d = 3 only the identity
matrix is Hermitian. Thus, when computing the coefficients of a density matrix in these bases, as we do next, the
Hermitian conjugation notation has to be retained. Note that the very last assertion in Corollary 1 corresponds to
the usualinequality relatingthe L magnitude ofa FouriertransformandtheL magnitiude ofthe originalfunction.
2 2
Corollary 1 (i)Thematrixelements ofa d ddensity matrixρinthe differentbasesarerelatedby(s)=F (a).
∗
(ii) s =1, s =ηjks and 1(F s) ×=ρ 0. ·
0,0 d−j,d−k ∗j,k d · j,0 j,j ≥
(iii) s 2 =d ρ 2 and s 2 ρ 2 1/√d.
j,k| j,k| | j,k| j,k| j,k| j,k| j,k| ≥
Proof: We expand an arbitrary dqensity matrixqin the two bases:
P P P P
1
ρ = a A = s S
j,k j,k j,k j,k
d
j,k j,k
X X
where aj,k = Tr A†j,kρ gives aj,k = ρj,j+k, using addition modd, and sj,k = Tr Sj†,kρ . Then from (7), sj,k =
dr=−01F∗(j,r)ar,(cid:16)k, whic(cid:17)h proves (i). Note that we have to include a complex conj(cid:16)ugatio(cid:17)n in the formula which is
unnecessary in the d = 2 case, since the Hadamard matrix has real entries. The assertions in (ii) follow from the
P
definitions and from the fact that ρ is Hermitian with trace equal to one. Finally the relations in (iii) follow from
Tr (s)†(s) = Tr (a)†(F∗)†(F∗)(a) = dTr (a)†(a) = dTr ρ2 , and from Tr ρ2 = kλ2k ≥ k1/d2 = 1/d,
whe(cid:16)re the λ(cid:17)k are th(cid:16)e non-negative eige(cid:17)nvalues o(cid:16)f the den(cid:17)sity ρ. (cid:0) (cid:1) (cid:0) (cid:1) P P
NowletN =d d d withd 2andwiththeorderofmultiplicationfixedthroughoutthediscussion. We
1 2 b i
use the underlying×and×fi·x·e·d×tensor prod≥uct representation of H[N] to define the sets of computational and adjusted
bases E[N] and A[N] for N N matrices as
×
E[N] = E(1) E(b) and A[N] = A(1) A(b) ,
j,k j1,k1 ⊗···⊗ jb,kb j,k j1,k1 ⊗···⊗ jb,kb
wherej andk correspondtotheirb-tuplesandthesuperscriptsinparenthesesidentifythecorrespondingd . Itfollows
i
that A[N] =E[N] where the addition of the indices is defined by
j,k j,j k
⊕
j k (j +k mod d ,...,j +k mod d ). (9)
1 1 1 b b b
⊕ ≡
The corresponding set of spin matrices S[N] is then defined by S[N] =F[N] A[N] or
(cid:0) (cid:1) (cid:0) (cid:1)
N 1
S[N] = − F[N](j,r)A[N]
j,k r,k
r=0
X
where F[N] =F(1) F(b) is the usual tensor product of the Fourier transforms F(k) which depend on d . Since
k
⊗···⊗
we will be taking powers of the η’s, we will use subscripts to denote the dependency of η on d : η = exp(2πi/d ).
k k k
It is easy to show that an equivalent definition of S[N] is given by
b
S[N] = F(i)A(i) . (10)
j,k
Oi=1(cid:16) (cid:17)ji,ki
4
Linearity again implies that if ρ[N] is a density matrix on the N N Hilbert space H[N] with
×
1
ρ[N] = a[N]A[N] = s[N]S[N],
j,k j,k N j,k j,k
j,k j,k
X X
then
s[N] =F [N] a[N] (11)
∗
·
(cid:16) (cid:17) (cid:16) (cid:17)
and a[N] =ρ[N] . Thus we have two different representations for a density matrix ρ[N], and both of them depend on
j,k j.j k
⊕
the underlying tensor product representation of H[N].
IV. SIGMA VARIATIONS
A fully separable density matrix can be represented as a convex combination of tensor products of pure states
or trace one projections, and we need to represent such d d projections in a systematic fashion in the spin basis.
×
(All projections in this paper are trace one projections.) In the Appendix we show how all trace one projections for
d=3 can be represented in a form completely analogous to the d=2 case, but for our immediate purposes we only
need to characterize a subclass. The motivation is given by writing the particular d = 2 projections 1(σ σ ) as
2 0± k
P (r) = 1 (( 1)rσ )m, where r = 0 or r = 1. Then P (r) is the average of the cyclic subgroup generated by
k m=0 − k k
( 1)rσ , and since (( 1)rσ )2 = σ , the key property P (r) P (r) = P (r) reduces to an exercise in group theory.
− k P − k 0 k · k k
The generalizationof this idea to arbitrary d is immediate, and we first treat the case when d is prime.
Proposition 2 Let d=p 2 be prime. Let u=(j,k)=(0,0) denote the index of a spin matrix S , and let r be
u
≥ 6
an integer. Then if p>2, the matrix
p 1
1 −
P (r) (ηrS )m (12)
u u
≡ p
m=0
X
is a projection with unit trace. The assertion is also valid for p = 2 provided iS is used in lieu of S throughout
u u
when u=(1,1).
Proof: A matrix P is a pure state or a trace one projection if it is Hermitian, has trace 1, and P2 = P. First,
(ηrS )m isproportionaltoS ;consequently,itcannotbeproportionaltoS for0<m<p. Therefore,onlyS
u mj,mk 0,0 0,0
contributes to the trace of P(u,r), confirming the trace condition. Using (8) it follows that ((ηrSu)m)† =(ηrSu)p−m
p(p−1)
and that when p is odd (ηrSu)m(ηrSu)p−m = (Su)p = ηjk 2 S0,0 = S0,0. Thus P (u,r) is Hermitian. The
p(p−1)
verification that P (u,r)2 = P (u,r) follows from an easy(cid:0)com(cid:1)putation. The assertion that ηjk 2 = 1 fails for
prime p only when p=2 and j =k =1. Thus, the reintroductionof i and of σ =iS is required to complete the
y 1,1
− (cid:0) (cid:1)
proof.
As an example of the notation, it is easy to check that if k =0, then P (r) is one of the diagonalprojections E .
j,0 i,i
Other projections are less sparse, however. For example, when d = 3 and k = 0, P (r) has no zero entries in the
j,k
6
computational basis representation.
In the preceding proof, we exploited the fact that for d an odd prime the powers of each matrix S ,u = (0,0),
u
6
form a cyclic subgroup of order d. When d is not prime we can get analogous results using a similar proof, but there
are restrictions on the indices that arise since the coefficient of the identity matrix in (8) when m = d need not be
unity. In Proposition 2 this led to the introduction of the factor i=exp(πi/2) when d=2, and that modification is
a special case of a more general situation.
Proposition 3 Suppose d is composite. Let u= (j,k) be (0,1), (1,0) or else an index such that j =0 and k = 0
6 6
d 1
have no common factors. Suppose d is odd or j k is even. Then if r is an integer, P (r) = 1 − (ηrS )m is a
· u d u
m=0
projection with unit trace. If d is even and j ·k is odd, then Pu(r) = d1 dm−=10(αηrSu)m is a projePction with unit
trace, where α=eπi/d.
Proof: Suppose (ηrS )m or (αηrS )m is proportional to S for 0 < mP< d, so that mj = rd and mk = sd for
u u 0,0
some integers r and s. Since j and k are relatively prime, there are integers a and b such that aj+bk = 1 [20], and
it follows that m=ard+bsd=(ar+bs)d, contradicting m<d. Thus P (r) has trace one. Using (8) when m=d,
u
5
we find that the coefficient of S is one in the first case, while in the second case the extra factor of αd = ( 1) is
0,0
−
necessary to make the overall coefficient equal to one. In both cases it follows from that key result as in Proposition
2 that P2(r)=P (r) and that P (r) is Hermitian, completing the proof.
u u u
An important relationship between these subgroup projections and the generating spin matrix follows from the
definitions.
Corollary 2 For any integer t 0 and any d 2,
≥ ≥
d 1
−
(ηrS )t = η mtP (m+r), (13)
u − u
m=0
X
subject to the usual caveat about α. In particular, S0,0 = dm−=10Pu(m+r)
Proof: dm−=10η−mtPu(m+r)= d1 k(ηrSu)k mη−mtηPmk =(ηrSu)t, as required.
Next consider a Hilbert space that is the direct product of b Hilbert spaces with dimensions d ,...,d . Projections
1 b
P P P
intheconstituentd dimensionalspacesalsodefineprojectionsintensorproductspaces,andtheproofofthefollowing
i
is immediate. As before, we let the superscript k denote the dependence on d .
k
Corollary 3 Let N = d1×d2 ×···×db and let H[N] = ba=1H[da] be an N dimensional Hilbert space. Let u
denote a b-dimensional vector of index pairs u =(j ,k ) where 0 j ,k d 1, and let r =(r ,...,r ) where the
i i i i i i 1 b
ri are integers. Then if the Pu(kk)(rk) are trace one projectionsNon H≤[dk], ≤ −
b
P (r) = P(k)(r )
u uk k
k=1
O
is a trace one projection on H[N], provided αS is used in place of S when d is even and u = (j,k) with j k odd.
u u
·
Furthermore, if η(r) b ηrk and t is a non-negative integer,
≡ k=1 k
Q t d1−1 db−1 b
η(r)Su[N] = ... ηk−lktPu(kk)(lk+rk), (14)
(cid:16) (cid:17) lX1=0 lXb=0Ok=1
and in particular S[N] =d1−1...db−1 b P(k)(l +r ).
0,0 k=1 uk k k
l1=0 lb=0
InordertoshowseparabPilityresPultsNforWernerdensities,weneedtoidentifyaspecialclassoffullyseparabledensity
matrices in the tensor product space H[N](d) of n d-dimensional Hilbert spaces, where N = dn. This approach is
motivated by results in [16] and is our final variation on the Pauli σ-matrices.
Proposition 4 Let d 2 and let u(n) = (u ,...,u ) and r(n) = (r ,...,r ) denote vectors of indices and values
1 n 1 n
≥
asdefined inthe precedingpropositions. ThenprovidedαS isusedin placeofS whendis evenandu=(j,k) with
u u
j k odd,
·
d 1
ρ u(n),r(n) = 1 (S S )+ − ((ηr1S )m (ηrnS )m)
dn 0,0⊗···⊗ 0,0 u1 ⊗···⊗ un !
(cid:16) (cid:17) mX=1
is a fully separable density matrix on H[N](d).
Proof: The assertion is true for n = 1 and suppose it holds for n. Let u(n+1) and r(n+1) be given index and
parameter vectors. Since we require only the nth and (n+1)st indices in the proof, we leave the other indices fixed
′ ′
and implicit and let ρ(u ,r ) denote ρ u(n),r(n) . By the induction hypothesis
n n
(cid:0)d 1 (cid:1)
1 −
ρ(u ,r +s) P(n+1)(r s)
d n n ⊗ un+1 n+1−
s=0
X
is fully separable. Multiplying out and collecting terms produces expressions of the form
d 1
1 −
[(ηr1Su1)m1 ⊗···⊗(ηrnSun)m1]⊗(ηrn+1Sun+1)m2d ηs(m1−m2).
s=0
X
By the same analysis used earlier, terms with m = m have an overall coefficient of 1 while all other terms have
1 2
coefficient 0, and that completes the proof of the induction step.
6
V. APPLICATIONS
We now have the tools to prove a sufficient condition for full and partial separability which extend the results in
[16]. This is a general sufficient condition for full or partial separability, and the results in [14] and [15] for N = 2n
and N = 3n respectively showing the existence of a neighborhood of the weighted identity in which every density is
fully separable follow as corollaries. As usual H[N] will denote an N dimensional Hilbert space which can be written
asatensorproduct: H[N] =H[d1] H[db],wherethe H[dk] aredk-dimensionalspacesandN =d1 d2 db.
⊗···⊗ × ×···×
We define D (d1,...,db) and refer to H[d1] H[db] as the D tensor product version of H[N]. Since H[N]
≡ ⊗···⊗
may be represented as a tensor product space in different ways, the kind of separability to be discussed depends on
the representation. For example, if N = 3n and H[N] is represented as the tensor product of n three-dimensional
spaces,we are discussing full separability. If subsets of the trits aretakentogether andrepresentedin 3k-dimensional
spaces,wearediscussingthecorrespondingpartialseparability. ByvirtueofLemma1,weknowthatthefundamental
mathematics involveddoesn’t depend on the order in which the tensor products are taken or which trits are grouped
together.
In expressing the condition of the theorem, we use the D spin coefficients to introduce an L norm on the space
1
of N N densities, and we will refer to that hereafter as the D spin norm and to the related separability as D
×
separability.
Theorem 1 Let H[N] denote an N-dimensional Hilbert space with N = d d d . Suppose H[N] =
1 2 b
H[d1] H[db], where the H[dk] are dk-dimensional Hilbert spaces. If ρ is a×densi×ty·m··a×trix on H[N], then ρ is
⊗···⊗
D (d ,...,d ) separable provided
1 b
≡
ρ s[N] 1, (15)
k k1,D ≡ j,k ≤
(j,kX)6=(0,0)(cid:12)(cid:12) (cid:12)(cid:12)
(cid:12) (cid:12)
where ρ has the spin representation 1 s[N]S[N] defined in term of the D tensor product: S[N] = b S(i) . It
N j,k j,k j,k j,k i=1 ji,ki
follows that in the set of density matrices on H[N] there is a neighborhood relative to D of the random state 1S[N]
P N N 0,0
in which every density matrix is D separable.
Proof: Ifd isprimeorj andk arerelativelyprime,thefactorS(i) canbewrittenasaweightedsumofprojections
i i i ji,ki
asinCorollary2. Ifd is compositeandthe indicesj andk arenotrelativelyprime,thenuptoafactorofηti,S(i)
i i i i ji,ki
s
can be written as ηrS(i) for some u= ¯j ,k¯ with ¯j and k¯ relatively prime, and thus S(i) can also be written
i u i i i i ji,ki
as a weighted sum(cid:16)of pro(cid:17)jections. Now si(cid:0)nce ρ(cid:1)is a density, either S[N] is Hermitian and thus s[N] is real, or S[N]
j,k j,k j,k
appearsin apair s[N]S[N]+s∗[N] S[N] †. In the secondcasewe use (14)in Corollary3andthe preceding comments
j,k j,k j,k j,k
to collect the various factors of ηt(cid:16)i toge(cid:17)ther and obtain
i
s[jN,k]Sj[N,k]+s∗j,[kN] Sj[N,k] † =d1−1...db−1 b Pu(kk)(lk+rk) βj,ks[jN,k]η∗(l)+βj∗,ks∗j,[kN]η(l)
(cid:16) (cid:17) lX1=0 lXb=0Ok=1 n o
= s[N] d1−1...db−1 b P(k)(l +r ) exp(iθ )η (l)+exp( iθ )η(l)
j,k uk k k { j,k ∗ − j,k }
(cid:12) (cid:12)lX1=0 lXb=0Ok=1
(cid:12) (cid:12)
(cid:12) (cid:12)
whereθ denotesthephaseofβ s[N] andl denotestheb-vectorwithcomponentsl . Thecaveatthatα S isinthe
j,k j,k j,k k i u
projections P(i)(l +r ) in lieu of S when d is even and u=(j ,k ) with j k odd applies throughoutthe proofand
uk k k u i i i i i
will not be explicitly cited. Since α has magnitude 1, only the phase factor will be affected. Using the last assertion
i
in Corollary 3, we can write s[jN,k] S0[N,0]+ 12 s[jN,k]Sj[N,k]+s∗j,[kN] Sj[N,k] † as
(cid:12) (cid:12) (cid:18) (cid:16) (cid:17) (cid:19)
(cid:12) (cid:12)
s(cid:12)[N] d(cid:12)1−1...db−1 b P (l +r ) 1+cos(θ arg(η(l))) .
j,k uk k k { j,k− }
(cid:12) (cid:12)lX1=0 lXb=0Ok=1
(cid:12) (cid:12)
(cid:12) (cid:12)
Sincetheexpressioninbracketsisnon-negative,theright-handsideisanon-negativemultipleofaD-separabledensity.
In the case when S[N] is Hermitian we derive the same expressionwith the same conclusion. It follows that ρ can be
j,k
7
writtenasaconvexcombinationoffullyseparabledensitiesplustheresidualterm 1 s[N] 1S[N]. The
− (j,k)=(0,0) j,k N 0,0
6
hypothesis guarantees that the coefficient of N1S0[N,0] is non-negative, and that com(cid:16)pletPes the proof(cid:12)(cid:12)of D(cid:12)(cid:12)(cid:17)-separability.
(cid:12) (cid:12)
As another application of the machinery, we can prove for prime p that the necessary condition s 1+pn−1 −1
≤
is sufficient for full separability of the generalized Werner density matrix W[N](s)= 1 s I +s τ. We have N =pn,
N− · (cid:0) (cid:1)
I is the identity, τ is the projection defined by the state ψ[N] = 1 p−1 k˜ and k˜ denotes the n-long repeated
√p k=0
index k...k. Given this special structure we find (cid:12) (cid:11) P (cid:12)(cid:12) E
(cid:12) (cid:12)
p 1p 1 p 1p 1
W[N](s) = 1−sI + s − − ˜j k˜ = 1−sI + s − − A[N]
pn p pn p ˜j,k˜
Xj=0Xk=0(cid:12)(cid:12) (cid:11)D (cid:12)(cid:12) Xj=0kX=0
(cid:12)
where we have used the modular vector addition defined in (9). Computing the spin coefficients gives s = 1,
0,0
s =0 if m is not a k˜ with 0 k<p, and otherwise
j,m
≤
s
s = F (j,r) δ(r,Ind)
j,k˜ ∗ p
r
X
where Ind= r˜:0 r <p . Using the dot product of the index vectors j r = j r mod p,
k k
{ ≤ } •
p 1 P
2πi s − 2πi
s = exp − (j r) a = 1+ exp − (j r˜) .
j,k˜ r (cid:18) p • (cid:19) r,k˜ p r=1 (cid:18) p • (cid:19)!
X X
Let Ind(p,n)={j : Nr=−01jr =0mod p}. Then it’s easy to check that sj,k˜ = s if and only if j is in Ind(p,n) and
that there are exactly pn 1 such indices. All other s equal zero, and we can write W[N](s) in the spin basis as
P − j,k˜
p 1
W[N](s)= 1−sS[N]+s − S[N]. (16)
pn 0,0 j,k˜
j Ind(p,n)k=0
∈ X X
Theorem 2 Let p be prime and N = pn. Then the generalized Werner density matrix W[N](s) is fully separable
on H[n](p) if and only if s 1+pn−1 −1.
≤
Proof: As shown above, nec(cid:0)essarily s(cid:1) 1+pn−1 −1. Checking the preceding derivation, note that
≤
(cid:0) (cid:1)
p 1 p 1
1 − E = 1 − A[N] = S[N]
p ˜j,˜j p ˜j,˜0 j,0
j=0 j=0 j Ind(p,n)
X X ∈ X
is a sum of fully separable projections. Taking s= 1+pn−1 −1 we can write W[n](s) as
(cid:0) (cid:1)
p 1 p 1
1 1 − 1 −
W[n](s) = E + S + S .
1+pn−1 pXj=0 ˜j,˜j j∈IXnd(p,n)pn ˜0,˜0 kX=1 j,k˜!
For each k =0, Ind(p,n) is mapped in a one-to-one manner onto itself by j kj where (kj) =kj mod p. Thus
6 → r r
p 1 p 1
1 1 − 1 −
W[n](s)= E + S + S . (17)
1+pn−1 p j=0 ˜j,˜j j Ind(p,n) pn ˜0,˜0 k=1 kj,k˜!!
X ∈ X X
But since
(S )k (S )k = ηk jiS = S
j1,1 ⊗···⊗ jn,1 kj,k˜ kj,k˜
P
for j in Ind(p,n), each j-sum in (17) is fully separable by Proposition 4, completing the proof.
8
It follows for the Werner densities that at the extreme value s = 1+pn−1 −1, (j,k)=(0,0) s[jN,k] = p1+1p−−p(−nn−1),
6
where the coefficients are based on the decomposition D = (p,...,p). When p = n = 2, that(cid:12)valu(cid:12)e is 1, showing
(cid:0) (cid:1) P (cid:12) (cid:12)
that the global bound of Theorem 1 is attained. However, for larger n and prime p 2 the co(cid:12)ndit(cid:12)ion ρ 1
≥ k k1,D ≤
is too strong for that class, and the special structure of the Werner densities allowed a more refined analysis of
D =(p,...,p)-separability.
It was shown in the qubit case in [16] that for each n and given ǫ > 0, there exists a D = (2,...,2)-inseparable
density on H[2n] which has ρ < 1+ǫ. Thus, for each fixed n the sufficient condition of Theorem 1 is the best
k k1,D
possible for full separability of qubits. We conjecture that the same is true in general: given any separability vector
D and ǫ>0 there exists a D-inseparable density ρ with ρ <1+ǫ.
k k1,D
ACKNOWLEDGMENTS
A. O. Pittenger gratefully acknowledges the hospitality of the Centre for Quantum Computation at Oxford Uni-
versityand supportfromUMBC andthe NationalSecurity Agency. M. H. Rubin wishes to thank the Office of Naval
Research and the National Security Agency for support of this work.
VI. APPENDIX
By emphasizing selected properties of projections for d = 2, we can obtain a representation of all (trace one)
projections in spin notation for d > 2. We concentrate on d = 3. To motivate the approach, recall from (5) that
when d = 2, m = (1)ρ +( 1)ρ , so that this particular spin coordinate is a convex combination of (+1) and
3 0,0 1,1
−
( 1), another way of stating the well-knowncorrespondence between m , the coefficient of σ , and the diagonalof ρ.
3 z
−
If ρ is also a projection, then in the computational coordinates, ρ = b b exp(i(ϕ ϕ )), so that fixing m fixes
j,k j k j k 3
ρ =b2 and ρ =b2, and only the phase factor θ = ϕ ϕ is unspecified. Using th−e change of basis formula, the
0,0 0 1,1 1 0− 1
two remaining spin coefficients of a projection with prescribed m are thus given in terms of the parameter θ by
3
m s 1 1 b b eiθ
1 = 0,1 = 0 1 ,
im s 1 1 b b e iθ
2 1,1 0 1 −
(cid:18)− (cid:19) (cid:18) (cid:19) (cid:18) − (cid:19)(cid:18) (cid:19)
where 0 θ <2π. If we let t denote the value of s when θ =0, we can rewrite the preceding equation as
k k,1
≤
s 1 1 1 eiθ 0 1 1 t
0,1 == 0 .
s1,1 2 1 1 0 e−iθ 1 1 t1
(cid:18) (cid:19) (cid:18) − (cid:19)(cid:18) (cid:19)(cid:18) − (cid:19)(cid:18) (cid:19)
Making the obvious definitions, this gives~s=M (θ)~t, and we also find that
2
cos(θ) isin(θ)
M (θ) = = cos(θ)σ +isin(θ)σ .
2 isin(θ) cos(θ) 0 x
(cid:18) (cid:19)
The geometry of this result is that if 1<m <1, then the remaining spin coefficients in the projections associated
3
with m can be identified with the ra−nge of a one parameter family of invertible mappings M (θ) acting on~t and
3 2
{ }
are represented by the intersection of the surface of the Bloch sphere with a horizontal plane at height m .
3
The same pattern of results holds for d = 3. Since s = s , the diagonal of a given ρ is in one-to-one corre-
2,0 ∗1,0
spondence with s via the equation s =ρ (1)+ρ η2 +ρ (η). That is, s is a convex combinationof the
1,0 1,0 0,0 1,1 2,2 1,0
vertices of an equilateral triangle in the complex plane and thus uniquely corresponds to the weights of the vertices,
(cid:0) (cid:1)
weights which are the entries of the diagonal of ρ. For larger values of d, the geometry is more complicated. For
example if d = 4, the diagonal of a given ρ corresponds to two spin coefficients: 1 s +1 and s which is
2,0 1,0
− ≤ ≤
restricted to a rectangle in the complex plane with vertices (1+s )/2 i(1 s )/2. In general the diagonal
2,0 2,0
± ± −
of a density matrix ρ corresponds to d/2 spin coefficients s , j = 0, when d is even and (d 1)/2 spin coefficients
j,0
6 −
when d is odd.
Once s is fixed in the d = 3 case, there are three complex parameters remaining to be specified: s ,s ,
1,0 0,1 1,1
and s , since the other four spin coefficients are forced by the restriction s = ηjks . If ρ is a projection,
2,1 3 j,3 k ∗j,k
− −
s 2 =3 ρ 2 =3, and thus s 2+ s 2 =1, tempting one to look for an analogue of the Bloch sphere
j,k j,k 2,0 k,1
| | | | | | | |
to represent all densities. However, the normalization arising from tr ρ2 = 1 is only a necessary condition on the
P P P
parameters,andexamples show it’s not sufficient. (See also [17].) Insteadwe follow the d=2 paradigmand describe
(cid:0) (cid:1)
9
trace one projections associated with a fixed s . If ρ is such a projection, then in the computational coordinates
1,0
ρ= u u, where u denotes a normalized three vector with uk =bkeiϕk and bk 2 =1. Fixing s1,0 fixes the bk’s,
| ih | | i | |
and it follows from Corollary 1 and the structure of ρ that
P
s0,1 1 1 1 eiθ0 0 0 b0b1
s1,1 = 1 η2 η 0 eiθ1 0 b1b2
s2,1 1 η η2 0 0 eiθ2 b2b0
where θ = ϕ ϕ with addition modulo 2π and with the normalization θ = 0mod 2π. Again letting t
k k k+1 k k
−
denotethevalueofs whentheθ ’sarechosentobezero,thistimeweobtainatwoparameterfamilyofprojections
k,1 k
associated with a given value of s . Letting ~s denote the column vector of paPrameters, ~t the column vector with
1,0
components t , and θ the 3 vector of phase parameters, we have~s=M (θ)~t, where
k 3
−
1 1 1 eiθ0 0 0 1 1 1 2
1
M3(θ)= 1 η2 η 0 eiθ1 0 1 η η2 = f(k,θ)S0,k. (18)
31 η η2 · 0 0 eiθ2 · 1 η2 η k=0
X
If θ+φ is defined as component-wise addition, then it is easy to check that M (θ) also defines an Abelian group
3
{ }
of invertible mappings,
M (θ) M (φ) = M (θ+φ),
3 3 3
·
giving the functional equation f(k,θ)f(j k,φ) = f(j,θ+φ) in analogy with the corresponding result when
k −
d=2. We have thus established a correspondence between all trace one projection matrices with given diagonal and
therangeofatwoparameterfamPilyofmappingsactingon~t. (WeareendebtedtoRasmusHansenforbringingtoour
attention [21], which contains an analysis of the geometry of the convex space of d = 3 densities. The pre-tranform
characterizationof the projections associated with a given diagonal is similar to the results derived here.)
In the d = 2 case the choices of m = 1 produce special cases of projections, and the same is true when d = 3.
z
If s is one of the extreme points 1, η±, or η2, then two of the b ’s equal zero and all of the s ’s equal zero.
1,0 k k,1
It follows that for r = 0, 1, and 2, 13 S0,0+ηrS1,0+(ηrS1,0)† is a trace one projection, and those are the three
subgroup projections P1,0(r). A degenehracy which has no analoigue in the d = 2 case occurs when s1,0 lies between
two extreme points on an edge. Then exactly one of the b ’s equals zero, and there is a one parameter family of
k
projections associated with s . The most interesting cases occur when s lies in the interior of the equilateral
1,0 1,0
triangle. In particular when s =0, the b ’s are equal to 1/√3, and by choosing the components of θ appropriately
1,0 k
from 0,2π/3,4π/3 we find the remaining subgroup projections P (r). Thus, our entire analysis of separability in
u
{ }
the d=3 case uses only the projections associated with the origin and with the vertices of the equilateral triangle.
[1] A.Einstein, B. Podolsky, and N. Rosen, Phys.Rev. 47, 777 (1935).
[2] A.Aspect, J. Dalibard, and G. Roger, Phys.Rev.Lett. 49, 1804 (1982).
[3] P.Shor, SIAMJ. Comput.26, no. 5, 1484 (1997).
[4] C. H. Bennett,et. al.,Phys. Rev.Lett. 70, 1895 (1993).
[5] C. H. Bennett,et. al.,PhysRev. A 59,1070 (1999).
[6] D. DiVincenzo, et. al., “Evidence for bound entangled states with negative partial transpose,” LANL quant/ph 9910026
v2, (1999).
[7] M. A.Nielsen, Phys.Rev. Lett. 83, 436 (1999).
[8] G. Vidal, “Entanglement monotones,” LANL quant-ph/9807077 (1998).
[9] W. Du¨r, J. I. Cirac, and R. Tarrach, Phys. Rev.Lett. 83, 3562 (1999).
[10] W. Dur, J. I. Cirac, “Classification of multi-qubit mixed states: separability and distillability properties,” LANL quant-
ph9911044 (1999)
[11] A.Peres, Phys.Rev. Lett. 77, 1413 (1996).
[12] M. Horodecki, P. Horodecki, and R. Horodecki, Phys. Lett.A 223, 1 (1996).
[13] R.Schack and C. M. Caves, quant-ph/9904109 v2.
[14] S.L. Braunstein, et. al., Phys.Rev.Lett. 83, 1054 (1999).
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